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PrologText Book: C.Kittel, "Introduction to Solid State Physics", 8th ed.,Wiley (2005) Website:http://ckw.phys.ncku.edu.twHomework submission:class@ckw.phys.ncku.edu.tw1. Crystal StructurePeriodic Arrays of AtomsFundamental Types of LatticesIndex System for Crystal PlanesSimple Crystal StructuresDirect Imaging of Atomic StructureCrystal Structure DataPeriodic Arrays of AtomsExperimental evidence of periodic structures: Integral idex numbers. X-ray diffraction (Laues theory).

Crystal building from identical blocks.

Lattice Translation VectorsCrystal = Lattice + BasisLattice = set of points given by

d = dimension of latticeai = translation vectors

If every pair of equivalent points in a crystal are related by (1), then ai are called primitive translation vectors.(1)Volume bounded by ai is called a cell.A crystal is invariant under the translation

Cell bounded by primitive translation vectors is called a primitive cell.

Basis:

Primitive Lattice CellRecapitulation:Parallelepiped defined by axes ai is called a cell.A cell must fill all space when subject to all possible lattice translations of the crystal.Parallelepiped defined by primitive axes ai is called a primitive cell.A crystal with 1 atom in its primitive cell is called a Bravais crystal.Characteristics of a primitive cell : Cell volume is minimal. Number of basis atoms is minimal. Contains exactly 1 lattice point.

Wigner-Seitz cellPrimitive cell centered at a lattice point and bounded by planes normal to and bisecting the lines joining the lattice point to its neighboring points.

Fundamental Types of LatticesOnly rotations Cn with n = 2, 3, 4 and 6 are compatible with the translational symmetry.Therere 32 crystallographic point groups (classes).Lattices with the same maximal point group are said to belong to the same crystal systems. Therere only 7 crystal systems in 3-D.Besides the primitive lattice (denoted by P or R ), some crystal systems may allow other centered lattices (denoted by C, A, F, or I ). Therere 14 Bravais lattices (lattice types) in 3-D and 5 in 2-D.2-D Bravais Lattices

= P = primary , b = C = base centered, v = I = body centered, f = F = face centerd

Cubic Lattices

Index System for Crystal PlanesMiller indices of a crystal plane:Express the intercepts of the plane with the crystal axes in units of lattice constants a1 , a2 , a3. Take the reciprocal of these numbers.Reduce them to integers of the same ratio: (h,k,l).Intercepts at 3a1, 2a2, and 2a3. Reciprocals are (1/3, 1/2, 1/2).Miller indices = (233).

Simple Crystal StructuresSodium Chloride (NaCl)

Cesium Chloride Structure

Close-Packed Structure

Hexagonal Closed-Packed

ABCABC fccABABAB hcpDiamond Structure

Cubic Zinc Sulfide Structure

Direct Imaging of Atomic StructureScanning Tunneling Microscopy (Chap 19)

(111) surface of fcc Pt at 4K.Nearest neighbor distance = 2.78A.Non-Ideal Crystal StructuresClosed packing:ABCABC fccABABAB hcpRandom StackingPolytypism: stacking with long period.E.g., ZnS has >150 polytypes; longest period =360 layers. SiC has >45 polytypes; longest period =594 layers.Cause: spiral steps due to dislocations in growth nucleus.Crystal Structure Data